Adding to Emilie's great answer
It's difficult to define what "stressful" is, because it varies from person to person (are designers more stressed than, say, surgeons?), so I'll just focus on the things that I think can make design different from other jobs.
Note: Graphic Design is a HUGE field. You can work in print, in web, in motion... you can do illustration, game design, development. All these specializations have different constrains and stress factors. I'll mention some general ones.
Stress Factor I: Requirements
One of the main "stressful factors", I think, is dealing with requirements from clients - if you are freelancing -, or from other departments - if you are working in a company. I haven't worked for an agency so I can't talk about that :)
You need to understand what the client wants, and deliver something that will work AND fit their expectations. I'll get a bit sensitive here and say that the closer your relationship with art is, the more difficult it can sometimes be to have to adjust your ideas. Most times, it actually challenges you and you end up with something better than you originally thought. Clients also come in all sizes and shapes, and I'm sure any freelancer can tell you a funny/terribly-frustrating anecdote.
Requirements from other departments can also be interesting to handle, as you usually need to work with several people - different expectations.
Stress Factor II: Deadlines and long nights
Deadlines are very well known, we've all had to 'suffer them' from school years. Working for a company, quite simple: You have to make the deadline, usually end of story. As stressful as in any other job.
For freelancers: Being that everything depends on your organizational skills, if you feel you are going to fail a deadline you will probably have to trade in some weekends and long nights. This gets easier as you get more ongoing remote work.
Stress Factor III: Art
I'll keep this one short. In most cases, you will be working for others. As simple as this sounds, if you are a creative person you will feel you have more to give but not enough time or mental energy to do it. If you are lucky, you will work on something that fills that need to think something new, so it is, in the end, quite relative.
Other than the previous, most of my designer friends (and me as well) affirm that they feel lucky to work in something they are passionate about. You can always love what you do, regardless of what it is. If you like design, I think right now is a great moment to be a designer. So many things happening, so much talent available and new ideas coming to life all the time. You will really enjoy meeting and working with other designers. So... it's not bad at all :)